= Fermi-window convolution of the grand potential
{title2=$\Phi(T,\mu)=\int_0^\infty\Phi_0(\epsilon)[-\partial_\epsilon n_F]d\epsilon$}
For noninteracting <fermions> with <density of states> $g$, define $\Phi_0(x)=-\int_0^x(x-y)g(y)dy$. The <grand potential> is $\int_0^\infty g(\epsilon)F(\epsilon)d\epsilon$, where $F=k_BT\ln(1-n_F)$ has derivative $F'=n_F$. Integrating by parts twice gives the title expression, since $\Phi_0(0)=\Phi'_0(0)=0$ and upper boundary terms vanish for ordinary polynomially growing <densities> of states. The <Fermi window> therefore thermally averages the zero-temperature <grand potential> as a function of trial <Fermi energy>.
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